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Chapter 9: Q. 9 (page 591)

Every nonzero complex number has exactly _______ distinct cube roots.

Short Answer

Expert verified

Every complex number has three cube roots of unity.

Step by step solution

01

Step 1. Given information

An incomplete statement is given as

Every nonzero complex number has exactly _______ distinct cube roots.

02

Step 2. Explanation

Let non zero complex number be w=rcos胃0+isin胃0

And the cube root for this complex number is

role="math" localid="1646734334742" zk=rncos胃0n+2k蟺n+isin胃0n+2k蟺n

Where k can be any positive integer.

For n=3we have k=0,1,2

As ktakes three values then we have exactly three cube roots.

Therefore every complex number has three cube roots of unity.

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Use the vectors in the figure at the right to graph each of the following vectors.

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